Center of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra .

Definition. The center of $\mathfrak{g}$ is

Basic properties.

• $Z(\mathfrak{g})$ is a Lie subalgebra and in fact an ideal of $\mathfrak{g}$.

• If $\operatorname{ad}:\mathfrak{g}\to \mathfrak{gl}(\mathfrak{g})$ is the adjoint representation , then

  $$Z(\mathfrak{g})=\ker(\operatorname{ad}).$$

• $\mathfrak{g}$ is abelian exactly when $Z(\mathfrak{g})=\mathfrak{g}$.

Context. The center measures how far $\mathfrak{g}$ is from being faithful under its own adjoint action. It is also the natural coefficient space for central extensions: a quotient by a central ideal is a quotient Lie algebra where “extra commuting directions” have been collapsed.